Very unbalanced Chess Positions Marco Ripà sPIqr Society, World Intelligence Network Rome, Italy e-mail: [email protected] Abstract: In this paper, we solve a few optimization problems regarding chess. It is well known that chess pieces have different values, and the standard scoring criteria assign 9 points to the queen (Q=9), 5 points to the rook (R=5), 3 points for both bishop and knight (B=3=N), while a pawn is evaluated only 1 point (P=1). Since kings cannot be taken, their value is undetermined; hence, we assume K=0. Our goal is to maximize the score gap between the White and the Black, under different constraints, letting the disadvantaged player checkmate his opponent, or end the game as a draw by stalemate. Keywords: Board games, Chess, 2-Person games, Two-Player games, Optimization problem, Checkmate, Stalemate, Zero-sum game, Pieces value, Game theory. 2010 Mathematics Subject Classification: 91A05, 91A24. 1 Introduction Chess is one of the oldest finite two-player zero-sum board games. FIDE standard rules state when a match can be declared as a draw, and one of them involves stalemate (see FIDE Laws of Chess, Article 9, rule 9.6 [3]). Stalemate occurs when the player on move is not able to perform any legal move, and at the same time is not threatened by check. Thus, stalemate can be forced by both players: we define it as "active" if the player moves without allowing his opponent to perform any legal move, and we call it "passive" if the player has no legal move available (it is an active stalemate for his opponent). In the present paper we study three different maximization problems, taking into account the final score on the board for both players. The aforementioned score is based on the standard chess piece relative value system [1], resumed in Potulates 1-5. Postulate 1. Pawn := (P) = 1 point; Knight := (N) = 3 points; Bishop := (B) = 3 points; Rook := (R) = 5 points; Queen := (Q) = 9 points; King := (K) = 0 points. Postulate 2. In the whole game (at any move), each player has a total score given by the sum of the values of all his pieces on the board. Postulate 3. The final score of each player is given by the total score by Postulate 2, when the last legal move of the match has been played. Postulate 4. The final move of a game is the one that produces checkmate or stalemate. Postulate 5. The final score difference between player White and player Black is given by ����� ����� �ℎ��� − ����� ����� �����, (1) while, at any intermediate move of the game, the score difference between player White and player Black is ����� �ℎ��� (������� ��������) − ����� ����� (������� ��������). (2) Obviously, the result from (2) is a non negative integer (natural numbers including zero) if and only if the White has no material disadvantage on the board at the current (half) move of the game. We add the following coinstraint to the 5 postulates stated above. Constraint 1. Considering any move of the game, for both players, ����e W − score B ≥ 0 is (always) true. The problems which we are going to solve in Section 2, under Constraint 1, are as follows: Problem 1. Which is the maximum value of (1) to let the Black win the game? Problem 2. Which is the maximum value of (1) to let the Black end the game with a passive stalemate? Problem 3. Which is the maximum value of (1) to let the Black end the game with an active stalemate? Problem 4. Which is the maximum value of ����� ����� �ℎ��� to let the Black end the game with an active stalemate? 2 Maximizing the negative score difference without losing the game We answer the questions introduced by Problems 1-4 (Section 1) using the well known PGN notation (Portable Game Notation [5]) in order to prove that any given final FEN (Forsyth- Edwards Notation [4]) position is legit (no matter if the same score can be achieved through a smaller number of moves and/or reaching a different position). Problem 1. Which is the maximum value of (1) to let the Black win the game? Lemma 1. A score of 103 points is the maximum score attainable in a chess match. Proof of Lemma 1. At the beginning of the game, there are 8 pawns on the board for each player: assuming that all of them are promoted to a queen and no other piece is lost, the total score is given by ������� ����e = 9 ∙ � + 2 ∙ � + 2 ∙ � + 2 ∙ � = 9 ∙ 9 + 2 ∙ 5 + 2 ∙ 3 + 2 ∙ 3 = 103. (3) Theorem 1. The maximum value of (1) to let the Black win the game is 102. Proof of Theorem 1. We invoke Lemma 1. Since a king cannot checkmate the other king by himself, he needs at least one pawn of the same color in order to win the match. It follows that the maximum value of (1) to let the Black win the game is given by ������� ����e minus P=1. Therefore, we complete the proof showing that a 102 score is really achievable in a standard game with only legal moves (FIDE rules), ending the match in a win for the Black. PGN (0 - 1) 1. a4 b5 2. axb5 a6 3. bxa6 Bb7 4. axb7 c6 5. bxa8=Q Qc8 6. b4 Qd8 7. b5 Qc7 8. b6 Qd8 9. b7 Na6 10. b8=Q c5 11. d4 f6 12. dxc5 Kf7 13. c6 Ke8 14. c7 Kf7 15. c8=Q Ke8 16. c4 Kf7 17. c5 Ke8 18. c6 Kf7 19. c7 Ke8 20. Qcxa6 Kf7 21. c8=Q Ke8 22. e4 Kf7 23. e5 Ke8 24. e6 Nh6 25. exd7+ Kf7 26. f4 g5 27. fxg5 Nf5 28. h4 h6 29. gxh6 Rg8 30. h7 Rg7 31. h8=Q Kg6 32. Kd2 Qe8 33. g4 Kf7 34. Kc3 Kg6 35. Kc4 Kf7 36. Kd5 Kg6 37. Na3 Kf7 38. Nc4 Kg6 39. dxe8=Q+ Rf7 40. Ba3 Bh6 41. Bd6 Bg7 42. h5+ Kg5 43. h6 Rf8 44. Qc5 Rg8 45. Qec6 Kg6 46. g5 Bf8 47. gxf6 Bg7 48. h7 Nd4 49. fxg7 Nc2 50. hxg8=Q Nd4 51. Qgd8 Kf5 52. g8=Q Nc2 53. Qg2 Nd4 54. Qhxd4 e6# £ The final position is shown in figure 1 (FEN: QQ1Q4/8/Q1QBp3/2QK1k2/2NQ4/8/6Q1/R2Q1BNR w - 0 55). Figure 1. Final position of a Black victory, despite of the 102 points difference. It is trivial to show how it is possible to change a few moves of the previous game to get an even 100 points final score difference and a loss for the White. As an example, let us modify moves 51 (et seq.) in order to checkmate the white king (103 points final score for the White) with one knight: PGN (0 - 1) 1. a4 b5 2. axb5 a6 3. bxa6 Bb7 4. axb7 c6 5. bxa8=Q Qc8 6. b4 Qd8 7. b5 Qc7 8. b6 Qd8 9. b7 Na6 10. b8=Q c5 11. d4 f6 12. dxc5 Kf7 13. c6 Ke8 14. c7 Kf7 15. c8=Q Ke8 16. c4 Kf7 17. c5 Ke8 18. c6 Kf7 19. c7 Ke8 20. Qcxa6 Kf7 21. c8=Q Ke8 22. e4 Kf7 23. e5 Ke8 24. e6 Nh6 25. exd7+ Kf7 26. f4 g5 27. fxg5 Nf5 28. h4 h6 29. gxh6 Rg8 30. h7 Rg7 31. h8=Q Kg6 32. Kd2 Qe8 33. g4 Kf7 34. Kc3 Kg6 35. Kc4 Kf7 36. Kd5 Kg6 37. Na3 Kf7 38. Nc4 Kg6 39. dxe8=Q+ Rf7 40. Ba3 Bh6 41. Bd6 Bg7 42. h5+ Kg5 43. h6 Rf8 44. Qc5 Rg8 45. Qec6 Kg6 46. g5 Bf8 47. gxf6 Bg7 48. h7 Nd4 49. fxg7 Nc2 50. hxg8=Q Nd4 51. Bxe7+ Kf5 52. Qgd8 Nb3 53. g8=Q Nc1 54. Rh4 Na2 55. Rd4 Nc1 56. Bd6 Na2 57. Qhh4 Nc3# Problem 2. Which is the maximum value of (1) to let the Black end the game with a passive stalemate? Theorem 2. The maximum value of (1) to let the Black end the game with a passive stalemate is 103 points. Proof of Theorem 2. Since Lemma 1 states that the maximum score achievable is 103 points, then we prove that 103 points is a valid score when an active stalemate by the White occurs [6]. PGN (1/2 - 1/2) 1. a4 b5 2. axb5 a6 3. bxa6 Bb7 4. axb7 c6 5. bxa8=Q d5 6. c4 e6 7. cxd5 Qd7 8. dxc6 Be7 9. c7 Kf8 10. c8=Q+ Qe8 11. b4 Nc6 12. b5 Nd8 13. b6 Nc6 14. b7 Nd8 15. b8=Q h5 16. d4 g6 17. d5 Rh7 18. d6 Rh8 19. d7 Rh7 20. e4 Nc6 21. d8=Q f5 22. exf5 Nh6 23. fxe6 Bf6 24. e7+ Kg8 25. f4 Qf8 26. e8=Q Bg7 27. f5 Kh8 28. f6 Qg8 29. f7 g5 30. f8=Q Nf7 31. h4 Nfe5 32. hxg5 Bf6 33. gxf6 Rg7 34. f7 Kh7 35. Qfa3 Qh8 36. f8=Q Rg5 37. g4 Qg7 38. gxh5 Qg6 39. Qfb4 Kg7 40. Qcxc6 Nf7 41. Qee2 Nd6 42. h6+ Kf7 43. h7 Rd5 44. Qcxd6 Kg7 45. Q6xd5 Qd6 46. Q5xd6 Kf7 47. h8=Q £ The final position is shown in figure 2 (FEN: QQ1Q3Q/5k2/3Q4/8/1Q6/Q7/4Q3/RNBQKBNR b KQ - 0 47). Figure 2. Final position of a passive Black stalemate, despite of a 103 points difference. Problem 3. Which is the maximum value of (1) to let the Black end the game with an active stalemate? Lemma 2.